The closing lemma for generalized recurrence in the plane
HTML articles powered by AMS MathViewer
- by Maria Lúcia Alvarenga Peixoto
- Trans. Amer. Math. Soc. 308 (1988), 143-158
- DOI: https://doi.org/10.1090/S0002-9947-1988-0946436-0
- PDF | Request permission
Abstract:
We prove a version of the Closing Lemma for ${C^r}$ vector fields on the plane, $r \geqslant 1$, and for a kind of recurrence obtained using the concept of prolongational limit sets. We call it generalized recurrence. Given a nonperiodic point $p$ in the generalized recurrent set we perturb the vector field in order to get a new vector field arbitrarily close to it, with a closed orbit through $p$.References
- Joseph Auslander, Generalized recurrence in dynamical systems, Contributions to Differential Equations 3 (1964), 65–74. MR 162238
- N. P. Bhatia and G. P. Szegö, Stability theory of dynamical systems, Die Grundlehren der mathematischen Wissenschaften, Band 161, Springer-Verlag, New York-Berlin, 1970. MR 0289890
- Freddy Dumortier, Singularities of vector fields on the plane, J. Differential Equations 23 (1977), no. 1, 53–106. MR 650816, DOI 10.1016/0022-0396(77)90136-X C. Gutierrez, On the ${C^r}$-closing lemma for flows on the torus ${T^2}$, Ergodic Theory and Dynamical Systems (to appear). —, A counterexample to a ${C^2}$ closing lemma (to appear).
- Ricardo Mañé, An ergodic closing lemma, Ann. of Math. (2) 116 (1982), no. 3, 503–540. MR 678479, DOI 10.2307/2007021
- P. Mendes, On stability of dynamical systems on open manifolds, J. Differential Equations 16 (1974), 144–167. MR 345137, DOI 10.1016/0022-0396(74)90031-X
- Janina Kotus, MichałKrych, and Zbigniew Nitecki, Global structural stability of flows on open surfaces, Mem. Amer. Math. Soc. 37 (1982), no. 261, v+108. MR 653093, DOI 10.1090/memo/0261
- Zbigniew Nitecki, Explosions in completely unstable flows. I. Preventing explosions, Trans. Amer. Math. Soc. 245 (1978), 43–61. MR 511399, DOI 10.1090/S0002-9947-1978-0511399-2
- Zbigniew Nitecki, Explosions in completely unstable flows. II. Some examples, Trans. Amer. Math. Soc. 245 (1978), 63–88. MR 511400, DOI 10.1090/S0002-9947-1978-0511400-6 J. Palis and W. de Melo, Geometric theory of dynamical systems, Springer-Verlag, Berlin and New York, 1983.
- M. M. Peixoto, Structural stability on two-dimensional manifolds, Topology 1 (1962), 101–120. MR 142859, DOI 10.1016/0040-9383(65)90018-2
- M. M. Peixoto, On an approximation theorem of Kupka and Smale, J. Differential Equations 3 (1967), 214–227. MR 209602, DOI 10.1016/0022-0396(67)90026-5
- M. M. Peixoto and C. C. Pugh, Structurally stable systems on open manifolds are never dense, Ann. of Math. (2) 87 (1968), 423–430. MR 224936, DOI 10.2307/1970713
- Charles C. Pugh, The closing lemma, Amer. J. Math. 89 (1967), 956–1009. MR 226669, DOI 10.2307/2373413
- Charles C. Pugh, An improved closing lemma and a general density theorem, Amer. J. Math. 89 (1967), 1010–1021. MR 226670, DOI 10.2307/2373414
- Charles Pugh, Against the $C^{2}$ closing lemma, J. Differential Equations 17 (1975), 435–443. MR 368079, DOI 10.1016/0022-0396(75)90054-6
- Charles C. Pugh and Clark Robinson, The $C^{1}$ closing lemma, including Hamiltonians, Ergodic Theory Dynam. Systems 3 (1983), no. 2, 261–313. MR 742228, DOI 10.1017/S0143385700001978 C. C. Pugh, The ${C^1}$-connecting lemma, a counterexample (preprint).
- Maria Lúcia Alvarenga Peixoto and Charles Chapman Pugh, The planar closing lemma for chain recurrence, Trans. Amer. Math. Soc. 341 (1994), no. 1, 173–192. MR 1107028, DOI 10.1090/S0002-9947-1994-1107028-9 J. Sotomayor, Generic one-parameter families of vector fields on two-dimensional manifolds, IHES, Extrait de Publications Mathematiques, 43.
- Taro Ura, Sur les courbes définies par les équations différentielles dans l’espace à $m$ dimensions, Ann. Sci. École Norm. Sup. (3) 70 (1953), 287–360 (French). MR 0064227 —, Sur le courant exterieur à une région invariante, Funkcial. Ekvac. 2 (1958), 143-200.
Bibliographic Information
- © Copyright 1988 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 308 (1988), 143-158
- MSC: Primary 58F10; Secondary 34D30
- DOI: https://doi.org/10.1090/S0002-9947-1988-0946436-0
- MathSciNet review: 946436